Construct Single-Hierarchical P/NBD Model for Online Retail Transaction Data

In this workbook we construct the non-hierarchical P/NBD models on the synthetic data with the longer timeframe.

1 Load and Construct Datasets

We start by modelling the P/NBD model using our synthetic datasets before we try to model real-life data.

1.1 Load Online Retail Data

We now want to load the online retail transaction data.

Code

customer_cohortdata_tbl <- read_rds("data/onlineretail_cohort_tbl.rds")
customer_cohortdata_tbl |> glimpse()

Rows: 5,852
Columns: 5
$ customer_id     <chr> "12346", "12347", "12348", "12349", "12350", "12351", …
$ cohort_qtr      <chr> "2010 Q1", "2010 Q4", "2010 Q3", "2010 Q2", "2011 Q1",…
$ cohort_ym       <chr> "2010 03", "2010 10", "2010 09", "2010 04", "2011 02",…
$ first_tnx_date  <date> 2010-03-02, 2010-10-31, 2010-09-27, 2010-04-29, 2011-…
$ total_tnx_count <int> 3, 8, 5, 3, 1, 1, 9, 2, 1, 2, 6, 2, 5, 10, 6, 4, 10, 2…

Code

customer_transactions_tbl <- read_rds("data/onlineretail_transactions_tbl.rds")
customer_transactions_tbl |> glimpse()

Rows: 36,658
Columns: 4
$ tnx_timestamp <dttm> 2009-12-01 07:45:00, 2009-12-01 07:45:59, 2009-12-01 09…
$ customer_id   <fct> 13085, 13085, 13078, 15362, 18102, 12682, 18087, 18087, …
$ invoice_id    <chr> "489434", "489435", "489436", "489437", "489438", "48943…
$ tnx_amount    <dbl> 505.30, 145.80, 630.33, 310.75, 2286.24, 372.30, 50.40, …

Code

customer_subset_id <- read_rds("data/onlineretail_customer_subset_ids.rds")
customer_subset_id |> glimpse()

 Factor w/ 5852 levels "13085","13078",..: 2 3 8 10 14 16 17 21 27 30 ...

1.2 Load Derived Data

Code

customer_summarystats_tbl <- read_rds("data/onlineretail_customer_summarystats_tbl.rds")

obs_fitdata_tbl   <- read_rds("data/onlineretail_obs_fitdata_tbl.rds")
obs_validdata_tbl <- read_rds("data/onlineretail_obs_validdata_tbl.rds")

customer_fit_stats_tbl <- obs_fitdata_tbl |>
  rename(x = tnx_count)

1.3 Load Subset Data

We also want to construct our data subsets for the purposes of speeding up our valuations.

Code

customer_fit_subset_tbl <- obs_fitdata_tbl |>
  filter(customer_id %in% customer_subset_id)

customer_fit_subset_tbl |> glimpse()

Rows: 1,000
Columns: 6
$ customer_id    <fct> 13078, 15362, 14110, 13758, 17592, 13767, 17238, 17700,…
$ first_tnx_date <dttm> 2009-12-01 09:05:59, 2009-12-01 09:08:00, 2009-12-01 0…
$ last_tnx_date  <dttm> 2010-11-30 13:17:59, 2010-09-17 10:36:59, 2010-09-30 1…
$ tnx_count      <dbl> 30, 1, 20, 6, 0, 35, 14, 9, 4, 9, 78, 2, 17, 5, 22, 2, …
$ t_x            <dbl> 52.02500, 41.43740, 43.32966, 51.72589, 0.00000, 52.034…
$ T_cal          <dbl> 52.08869, 52.08849, 52.08433, 52.08274, 52.07847, 52.07…

Code

customer_valid_subset_tbl <- obs_validdata_tbl |>
  filter(customer_id %in% customer_subset_id)

customer_valid_subset_tbl |> glimpse()

Rows: 1,000
Columns: 3
$ customer_id       <fct> 13078, 15362, 14110, 13758, 17592, 13767, 17238, 177…
$ tnx_count         <dbl> 26, 0, 12, 7, 0, 37, 17, 4, 4, 13, 48, 0, 13, 3, 16,…
$ tnx_last_interval <dbl> 52.950000, NA, 52.931151, 51.780853, NA, 53.089286, …

We now use these datasets to set the start and end dates for our various validation methods.

Code

dates_lst <- read_rds("data/onlineretail_simulation_dates.rds")

use_fit_start_date <- dates_lst$use_fit_start_date
use_fit_end_date   <- dates_lst$use_fit_end_date

use_valid_start_date <- dates_lst$use_valid_start_date
use_valid_end_date   <- dates_lst$use_valid_end_date

We now split out the transaction data into fit and validation datasets.

Code

customer_fit_transactions_tbl <- customer_transactions_tbl |>
  filter(
    customer_id %in% customer_subset_id,
    tnx_timestamp >= use_fit_start_date,
    tnx_timestamp <= use_fit_end_date
    )
  
customer_fit_transactions_tbl |> glimpse()

Rows: 4,164
Columns: 4
$ tnx_timestamp <dttm> 2009-12-01 09:05:59, 2009-12-01 09:08:00, 2009-12-01 09…
$ customer_id   <fct> 13078, 15362, 14110, 13758, 17592, 13767, 17238, 17700, …
$ invoice_id    <chr> "489436", "489437", "489443", "489446", "489462", "48946…
$ tnx_amount    <dbl> 630.33, 310.75, 285.06, 996.10, 148.30, 1197.80, 251.10,…

Code

customer_valid_transactions_tbl <- customer_transactions_tbl |>
  filter(
    customer_id %in% customer_subset_id,
    tnx_timestamp >= use_valid_start_date,
    tnx_timestamp <= use_valid_end_date
    )
  
customer_valid_transactions_tbl |> glimpse()

Rows: 3,422
Columns: 4
$ tnx_timestamp <dttm> 2010-12-01 09:32:00, 2010-12-01 09:59:00, 2010-12-01 10…
$ customer_id   <fct> 15291, 16250, 12431, 13408, 13767, 12791, 17908, 16583, …
$ invoice_id    <chr> "536376", "536388", "536389", "536394", "536395", "53640…
$ tnx_amount    <dbl> 657.60, 452.28, 716.50, 2049.36, 1015.76, 355.20, 486.56…

Finally, we want to extract the first transaction for each customer, so we can add this data to assess our models.

Code

customer_initial_tnx_tbl <- customer_fit_transactions_tbl |>
  slice_min(n = 1, order_by = tnx_timestamp, by = customer_id)

customer_initial_tnx_tbl |> glimpse()

Rows: 1,002
Columns: 4
$ tnx_timestamp <dttm> 2009-12-01 09:05:59, 2009-12-01 09:08:00, 2009-12-01 09…
$ customer_id   <fct> 13078, 15362, 14110, 13758, 17592, 13767, 17238, 17700, …
$ invoice_id    <chr> "489436", "489437", "489443", "489446", "489462", "48946…
$ tnx_amount    <dbl> 630.33, 310.75, 285.06, 996.10, 148.30, 1197.80, 251.10,…

We now expand out these initial transactions so that we can append them to our simulations.

Code

sim_init_tbl <- customer_initial_tnx_tbl |>
  transmute(
    customer_id,
    draw_id       = list(1:n_sim),
    tnx_timestamp,
    tnx_amount
    ) |>
  unnest(draw_id)

sim_init_tbl |> glimpse()

Rows: 2,004,000
Columns: 4
$ customer_id   <fct> 13078, 13078, 13078, 13078, 13078, 13078, 13078, 13078, …
$ draw_id       <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1…
$ tnx_timestamp <dttm> 2009-12-01 09:05:59, 2009-12-01 09:05:59, 2009-12-01 09…
$ tnx_amount    <dbl> 630.33, 630.33, 630.33, 630.33, 630.33, 630.33, 630.33, …

Before we start on that, we set a few parameters for the workbook to organise our Stan code.

Code

stan_modeldir <- "stan_models"
stan_codedir  <-   "stan_code"

2 Fit First Hierarchical Lambda Model

Our first hierarchical model puts a hierarchical prior around the mean of our population \(\lambda\) - lambda_mn.

Once again we use a Gamma prior for it.

2.1 Compile and Fit Stan Model

We now compile this model using CmdStanR.

Code

pnbd_onehierlambda_stanmodel <- cmdstan_model(
  "stan_code/pnbd_onehier_lambda.stan",
  include_paths =   stan_codedir,
  pedantic      =           TRUE,
  dir           =  stan_modeldir
  )

We then use this compiled model with our data to produce a fit of the data.

Code

stan_modelname <- "pnbd_onlineretail_onehierlambda1"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed   <- stanfit_seed + 1

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")


stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    hier_lambda_mn_p1 = 0.25,
    hier_lambda_mn_p2 = 1,

    lambda_cv = 1.00,
    
    mu_mn     = 0.10,
    mu_cv     = 1.00,
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_onlineretail_onehierlambda1_stanfit <- pnbd_onehierlambda_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )
  
  pnbd_onlineretail_onehierlambda1_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  pnbd_onlineretail_onehierlambda1_stanfit <- read_rds(stanfit_object_file)
}

pnbd_onlineretail_onehierlambda1_stanfit$print()

  variable      mean    median    sd   mad        q5       q95 rhat ess_bulk
 lp__      -61819.94 -61819.00 73.92 74.13 -61943.93 -61700.89 1.00      722
 lambda_mn      0.13      0.13  0.00  0.00      0.12      0.13 1.00     1463
 lambda[1]      0.32      0.30  0.14  0.13      0.13      0.56 1.00     2183
 lambda[2]      0.52      0.51  0.09  0.09      0.38      0.68 1.00     2463
 lambda[3]      0.03      0.03  0.02  0.02      0.01      0.08 1.00     2032
 lambda[4]      1.43      1.42  0.15  0.15      1.19      1.69 1.00     1988
 lambda[5]      0.33      0.33  0.07  0.07      0.22      0.46 1.00     2349
 lambda[6]      0.25      0.24  0.06  0.06      0.15      0.36 1.00     2144
 lambda[7]      0.05      0.04  0.03  0.03      0.01      0.10 1.00     1830
 lambda[8]      0.38      0.37  0.09  0.08      0.25      0.53 1.00     2142
 ess_tail
     1105
     1593
     1016
     1425
     1141
     1379
     1296
     1136
     1049
     1385

 # showing 10 of 12720 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)

We have some basic HMC-based validity statistics we can check.

Code

pnbd_onlineretail_onehierlambda1_stanfit$cmdstan_diagnose()

Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehierlambda1-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehierlambda1-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehierlambda1-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehierlambda1-4.csvWarning: non-fatal error reading adaptation data


Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

2.2 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Code

parameter_subset <- c(
  "lambda_mn",
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_onlineretail_onehierlambda1_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We also check \(N_{eff}\) as a quick diagnostic of the fit.

Code

pnbd_onlineretail_onehierlambda1_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

Finally, we want to check out the energy diagnostic, which is often indicative of problems with the posterior mixing.

Code

pnbd_onlineretail_onehierlambda1_stanfit |>
  nuts_params() |>
  mcmc_nuts_energy(binwidth = 50)

2.3 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

Code

pnbd_stanfit <- pnbd_onlineretail_onehierlambda1_stanfit |>
  recover_types(customer_fit_stats_tbl)

pnbd_onlineretail_onehierlambda1_assess_data_lst <- run_model_assessment(
  model_stanfit    = pnbd_stanfit,
  insample_tbl     = customer_fit_subset_tbl,
  fit_label        = "pnbd_onlineretail_onehierlambda1",
  fit_end_dttm     = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm   = use_valid_end_date   |> as.POSIXct(),
  sim_seed         = 1110
  )

pnbd_onlineretail_onehierlambda1_assess_data_lst |> glimpse()

List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_onlineretail_onehierlambda1_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_onlineretail_onehierlambda1_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_onlineretail_onehierlambda1_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_onlineretail_onehierlambda1_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_onlineretail_onehierlambda1_assess_valid_simstats_tbl.rds"

2.3.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Code

simdata_tbl <- pnbd_onlineretail_onehierlambda1_assess_data_lst |>
  use_series(model_fit_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  bind_rows(sim_init_tbl) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_fit_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

2.3.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Code

simdata_tbl <- pnbd_onlineretail_onehierlambda1_assess_data_lst |>
  use_series(model_valid_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_valid_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

As for our short time frame data, overall our model is working well.

3 Fit Second Hierarchical Lambda Model

In this model, we are going with a broadly similar model but we are instead using a different mean for our hierarchical prior.

3.1 Fit Stan Model

We now want to fit the model to our data using this alternative prior for lambda_mn.

Code

stan_modelname <- "pnbd_onlineretail_onehierlambda2"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed   <- stanfit_seed + 1

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")


stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    hier_lambda_mn_p1 = 0.50,
    hier_lambda_mn_p2 = 1,

    lambda_cv = 1.00,
    
    mu_mn     = 0.10,
    mu_cv     = 1.00,
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_onlineretail_onehierlambda2_stanfit <- pnbd_onehierlambda_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )
  
  pnbd_onlineretail_onehierlambda2_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  pnbd_onlineretail_onehierlambda2_stanfit <- read_rds(stanfit_object_file)
}

pnbd_onlineretail_onehierlambda2_stanfit$print()

  variable      mean    median    sd   mad        q5       q95 rhat ess_bulk
 lp__      -61818.40 -61816.80 70.87 73.91 -61937.42 -61706.67 1.01      599
 lambda_mn      0.13      0.13  0.00  0.00      0.12      0.13 1.00     1689
 lambda[1]      0.32      0.30  0.14  0.14      0.13      0.57 1.00     1818
 lambda[2]      0.52      0.51  0.10  0.09      0.37      0.69 1.00     2395
 lambda[3]      0.03      0.03  0.02  0.02      0.01      0.08 1.00     2085
 lambda[4]      1.43      1.42  0.16  0.15      1.18      1.71 1.00     2764
 lambda[5]      0.33      0.33  0.07  0.07      0.22      0.47 1.00     2652
 lambda[6]      0.25      0.24  0.06  0.06      0.16      0.37 1.00     2042
 lambda[7]      0.05      0.04  0.03  0.03      0.01      0.11 1.00     1670
 lambda[8]      0.38      0.37  0.09  0.08      0.25      0.53 1.00     2027
 ess_tail
     1031
     1576
     1227
     1400
      987
     1518
     1369
     1490
     1133
     1199

 # showing 10 of 12720 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)

We have some basic HMC-based validity statistics we can check.

Code

pnbd_onlineretail_onehierlambda2_stanfit$cmdstan_diagnose()

Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehierlambda2-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehierlambda2-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehierlambda2-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehierlambda2-4.csvWarning: non-fatal error reading adaptation data


Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

3.2 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Code

parameter_subset <- c(
  "lambda_mn",
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_onlineretail_onehierlambda2_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We also check \(N_{eff}\) as a quick diagnostic of the fit.

Code

pnbd_onlineretail_onehierlambda2_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

Finally, we want to check out the energy diagnostic, which is often indicative of problems with the posterior mixing.

Code

pnbd_onlineretail_onehierlambda2_stanfit |>
  nuts_params() |>
  mcmc_nuts_energy(binwidth = 50)

3.3 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

Code

pnbd_stanfit <- pnbd_onlineretail_onehierlambda2_stanfit |>
  recover_types(customer_fit_stats_tbl)

pnbd_onlineretail_onehierlambda2_assess_data_lst <- run_model_assessment(
  model_stanfit    = pnbd_stanfit,
  insample_tbl     = customer_fit_subset_tbl,
  fit_label        = "pnbd_onlineretail_onehierlambda2",
  fit_end_dttm     = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm   = use_valid_end_date   |> as.POSIXct(),
  sim_seed         = 1120
  )

pnbd_onlineretail_onehierlambda2_assess_data_lst |> glimpse()

List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_onlineretail_onehierlambda2_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_onlineretail_onehierlambda2_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_onlineretail_onehierlambda2_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_onlineretail_onehierlambda2_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_onlineretail_onehierlambda2_assess_valid_simstats_tbl.rds"

3.3.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Code

simdata_tbl <- pnbd_onlineretail_onehierlambda2_assess_data_lst |>
  use_series(model_fit_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  bind_rows(sim_init_tbl) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_fit_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

3.3.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Code

simdata_tbl <- pnbd_onlineretail_onehierlambda2_assess_data_lst |>
  use_series(model_valid_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_valid_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

As for our short time frame data, overall our model is working well.

4 Fit First Hierarchical Mu Model

We now construct the same hierarchical model but based around mu_mn.

4.1 Compile and Fit Stan Model

We compile this model using CmdStanR.

Code

pnbd_onehiermu_stanmodel <- cmdstan_model(
  "stan_code/pnbd_onehier_mu.stan",
  include_paths =   stan_codedir,
  pedantic      =           TRUE,
  dir           =  stan_modeldir
  )

We then use this compiled model with our data to produce a fit of the data.

Code

stan_modelname <- "pnbd_onlineretail_onehiermu1"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed   <- stanfit_seed + 1

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")


stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    hier_mu_mn_p1 = 0.50,
    hier_mu_mn_p2 = 1.00,

    lambda_mn     = 0.25,
    lambda_cv     = 1.00,

    mu_cv         = 1.00

    )

if(!file_exists(stanfit_object_file)) {
  pnbd_onlineretail_onehiermu1_stanfit <- pnbd_onehiermu_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )
  
  pnbd_onlineretail_onehiermu1_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  pnbd_onlineretail_onehiermu1_stanfit <- read_rds(stanfit_object_file)
}

pnbd_onlineretail_onehiermu1_stanfit$print()

  variable      mean    median    sd   mad        q5       q95 rhat ess_bulk
 lp__      -57192.99 -57191.35 72.03 73.02 -57313.38 -57075.27 1.00      724
 mu_mn          0.00      0.00  0.00  0.00      0.00      0.01 1.03       93
 lambda[1]      0.40      0.38  0.18  0.17      0.16      0.72 1.00     1863
 lambda[2]      0.55      0.55  0.10  0.10      0.41      0.72 1.00     3014
 lambda[3]      0.04      0.03  0.03  0.02      0.01      0.09 1.00     1935
 lambda[4]      1.52      1.52  0.17  0.17      1.26      1.82 1.00     2502
 lambda[5]      0.36      0.35  0.08  0.08      0.23      0.50 1.00     2551
 lambda[6]      0.27      0.26  0.07  0.06      0.17      0.39 1.01     1904
 lambda[7]      0.05      0.05  0.03  0.03      0.02      0.11 1.00     2090
 lambda[8]      0.39      0.38  0.09  0.09      0.25      0.54 1.00     2470
 ess_tail
     1058
      219
     1130
     1705
      991
     1465
     1380
     1004
     1236
     1302

 # showing 10 of 12720 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)

We have some basic HMC-based validity statistics we can check.

Code

pnbd_onlineretail_onehiermu1_stanfit$cmdstan_diagnose()

Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehiermu1-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehiermu1-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehiermu1-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehiermu1-4.csvWarning: non-fatal error reading adaptation data


Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

4.2 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Code

parameter_subset <- c(
  "mu_mn",
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_onlineretail_onehiermu1_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We also check \(N_{eff}\) as a quick diagnostic of the fit.

Code

pnbd_onlineretail_onehiermu1_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

Finally, we want to check out the energy diagnostic, which is often indicative of problems with the posterior mixing.

Code

pnbd_onlineretail_onehiermu1_stanfit |>
  nuts_params() |>
  mcmc_nuts_energy(binwidth = 50)

4.3 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

Code

pnbd_stanfit <- pnbd_onlineretail_onehiermu1_stanfit |>
  recover_types(customer_fit_stats_tbl)

pnbd_onlineretail_onehiermu1_assess_data_lst <- run_model_assessment(
  model_stanfit    = pnbd_stanfit,
  insample_tbl     = customer_fit_subset_tbl,
  fit_label        = "pnbd_onlineretail_onehiermu1",
  fit_end_dttm     = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm   = use_valid_end_date   |> as.POSIXct(),
  sim_seed         = 1130
  )

pnbd_onlineretail_onehiermu1_assess_data_lst |> glimpse()

List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_onlineretail_onehiermu1_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_onlineretail_onehiermu1_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_onlineretail_onehiermu1_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_onlineretail_onehiermu1_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_onlineretail_onehiermu1_assess_valid_simstats_tbl.rds"

4.3.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Code

simdata_tbl <- pnbd_onlineretail_onehiermu1_assess_data_lst |>
  use_series(model_fit_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  bind_rows(sim_init_tbl) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_fit_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

4.3.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Code

simdata_tbl <- pnbd_onlineretail_onehiermu1_assess_data_lst |>
  use_series(model_valid_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_valid_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

As for our short time frame data, overall our model is working well.

5 Fit Second Hierarchical Mu Model

In this model, we are going with a broadly similar model but we are instead using a different mean for our hierarchical prior.

5.1 Fit Stan Model

We now want to fit the model to our data using this alternative prior for lambda_mn.

Code

stan_modelname <- "pnbd_onlineretail_onehiermu2"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed   <- stanfit_seed + 1

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")


stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    hier_mu_mn_p1 = 0.25,
    hier_mu_mn_p2 = 1.00,

    lambda_mn     = 0.25,
    lambda_cv     = 1.00,

    mu_cv         = 1.00
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_onlineretail_onehiermu2_stanfit <- pnbd_onehiermu_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )
  
  pnbd_onlineretail_onehiermu2_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  pnbd_onlineretail_onehiermu2_stanfit <- read_rds(stanfit_object_file)
}

pnbd_onlineretail_onehiermu2_stanfit$print()

  variable      mean    median    sd   mad        q5       q95 rhat ess_bulk
 lp__      -57192.41 -57194.20 72.96 71.91 -57315.44 -57069.79 1.00      729
 mu_mn          0.00      0.00  0.00  0.00      0.00      0.01 1.05       82
 lambda[1]      0.39      0.37  0.18  0.18      0.15      0.73 1.00     1990
 lambda[2]      0.55      0.55  0.10  0.11      0.40      0.73 1.00     1573
 lambda[3]      0.04      0.03  0.03  0.02      0.01      0.09 1.00     1956
 lambda[4]      1.52      1.51  0.17  0.17      1.26      1.79 1.00     2160
 lambda[5]      0.36      0.35  0.08  0.08      0.24      0.49 1.00     2103
 lambda[6]      0.27      0.26  0.07  0.07      0.16      0.39 1.00     2258
 lambda[7]      0.05      0.05  0.03  0.03      0.01      0.11 1.00     2101
 lambda[8]      0.39      0.38  0.09  0.09      0.26      0.53 1.00     2252
 ess_tail
     1053
      159
     1276
     1100
     1053
     1486
     1470
     1239
     1314
     1594

 # showing 10 of 12720 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)

We have some basic HMC-based validity statistics we can check.

Code

pnbd_onlineretail_onehiermu2_stanfit$cmdstan_diagnose()

Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehiermu2-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehiermu2-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehiermu2-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_onlineretail_onehiermu2-4.csvWarning: non-fatal error reading adaptation data


Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

5.2 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Code

parameter_subset <- c(
  "mu_mn",
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_onlineretail_onehiermu2_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We also check \(N_{eff}\) as a quick diagnostic of the fit.

Code

pnbd_onlineretail_onehiermu2_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

Finally, we want to check out the energy diagnostic, which is often indicative of problems with the posterior mixing.

Code

pnbd_onlineretail_onehiermu2_stanfit |>
  nuts_params() |>
  mcmc_nuts_energy(binwidth = 50)

5.3 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

Code

pnbd_stanfit <- pnbd_onlineretail_onehiermu2_stanfit |>
  recover_types(customer_fit_stats_tbl)

pnbd_onlineretail_onehiermu2_assess_data_lst <- run_model_assessment(
  model_stanfit    = pnbd_stanfit,
  insample_tbl     = customer_fit_subset_tbl,
  fit_label        = "pnbd_onlineretail_onehiermu2",
  fit_end_dttm     = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm   = use_valid_end_date   |> as.POSIXct(),
  sim_seed         = 1140
  )

pnbd_onlineretail_onehiermu2_assess_data_lst |> glimpse()

List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_onlineretail_onehiermu2_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_onlineretail_onehiermu2_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_onlineretail_onehiermu2_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_onlineretail_onehiermu2_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_onlineretail_onehiermu2_assess_valid_simstats_tbl.rds"

5.3.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Code

simdata_tbl <- pnbd_onlineretail_onehiermu2_assess_data_lst |>
  use_series(model_fit_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  bind_rows(sim_init_tbl) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_fit_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

5.3.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Code

simdata_tbl <- pnbd_onlineretail_onehiermu2_assess_data_lst |>
  use_series(model_valid_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_valid_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

As for our short time frame data, overall our model is working well.

6 Compare Model Outputs

We have looked at each of the models individually, but it is also worth looking at each of the models as a group.

We now want to combine both the fit and valid transaction sets to calculate the summary statistics for both.

Code

obs_summstats_tbl <- list(
    fit   = customer_fit_transactions_tbl,
    valid = customer_valid_transactions_tbl
    ) |>
  bind_rows(.id = "assess_type") |>
  group_by(assess_type) |>
  calculate_transaction_summary_statistics() |>
  pivot_longer(
    cols      = !assess_type,
    names_to  = "label",
    values_to = "obs_value"
    )

obs_summstats_tbl |> glimpse()

Rows: 16
Columns: 3
$ assess_type <chr> "fit", "fit", "fit", "fit", "fit", "fit", "fit", "fit", "v…
$ label       <chr> "p10", "p25", "p50", "p75", "p90", "p99", "total_count", "…
$ obs_value   <dbl> 1.000000, 1.000000, 2.000000, 5.000000, 9.000000, 27.01000…

Code

model_assess_transactions_tbl <- dir_ls("data", regexp = "pnbd_onlineretail_(fixed|one).*_assess_.*index") |>
  enframe(name = NULL, value = "file_path") |>
  mutate(
    model_label = str_replace(file_path, "data/pnbd_onlineretail_(.*?)_assess_.*", "\\1"),
    assess_type = if_else(str_detect(file_path, "_assess_fit_"), "fit", "valid"),
    
    assess_data = map(
      file_path, construct_model_assessment_data,
      
      .progress = "construct_assess_data"
      )
    ) |>
  select(model_label, assess_type, assess_data) |>
  unnest(assess_data)

model_assess_transactions_tbl |> glimpse()

Rows: 116,133,294
Columns: 6
$ model_label   <chr> "fixed1", "fixed1", "fixed1", "fixed1", "fixed1", "fixed…
$ assess_type   <chr> "fit", "fit", "fit", "fit", "fit", "fit", "fit", "fit", …
$ customer_id   <fct> 13078, 13078, 13078, 13078, 13078, 13078, 13078, 13078, …
$ draw_id       <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,…
$ tnx_timestamp <dttm> 2009-12-12 19:15:15, 2009-12-17 19:57:58, 2010-01-07 00…
$ tnx_amount    <dbl> 181.28, 8.86, 109.95, 50.98, 146.65, 182.68, 67.32, 42.0…

We now want to calculate the transaction statistics on this full dataset, for each separate draw.

Code

model_assess_tbl <- model_assess_transactions_tbl |>
  group_by(model_label, assess_type, draw_id) |>
  calculate_transaction_summary_statistics()

model_assess_tbl |> glimpse()

Rows: 32,000
Columns: 11
$ model_label <chr> "fixed1", "fixed1", "fixed1", "fixed1", "fixed1", "fixed1"…
$ assess_type <chr> "fit", "fit", "fit", "fit", "fit", "fit", "fit", "fit", "f…
$ draw_id     <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,…
$ p10         <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
$ p25         <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1…
$ p50         <dbl> 3.0, 3.0, 2.5, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0…
$ p75         <dbl> 6, 5, 5, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 6, 6, 6…
$ p90         <dbl> 10, 9, 10, 10, 10, 11, 11, 10, 10, 10, 10, 10, 10, 10, 11,…
$ p99         <dbl> 31.03, 26.10, 26.53, 28.59, 25.69, 29.00, 25.70, 24.60, 23…
$ total_count <int> 2965, 2838, 2863, 2973, 2937, 2992, 2809, 2913, 2916, 2912…
$ mean_count  <dbl> 4.958194, 4.441315, 4.418210, 4.630841, 4.647152, 5.020134…

We now combine all this data to create a number of different comparison plots for the various summary statistics.

Code

#! echo: TRUE

create_multiple_model_assessment_plot(
  obs_summstats_tbl, model_assess_tbl,
  "total_count", "Total Transactions"
  )

Code

create_multiple_model_assessment_plot(
  obs_summstats_tbl, model_assess_tbl,
  "mean_count", "Average Transactions per Customer"
  )

Code

create_multiple_model_assessment_plot(
  obs_summstats_tbl, model_assess_tbl,
  "p99", "99th Percentile Count"
  )

6.1 Write Assessment Data to Disk

We now want to save the assessment data to disk.

Code

model_assess_tbl |> write_rds("data/assess_data_pnbd_onlineretail_onehier_tbl.rds")

R Environment

Code

options(width = 120L)
sessioninfo::session_info()

─ Session info ───────────────────────────────────────────────────────────────────────────────────────────────────────
 setting  value
 version  R version 4.3.1 (2023-06-16)
 os       Ubuntu 22.04.3 LTS
 system   x86_64, linux-gnu
 ui       X11
 language (EN)
 collate  en_US.UTF-8
 ctype    en_US.UTF-8
 tz       Europe/Dublin
 date     2023-11-20
 pandoc   3.1.1 @ /usr/local/bin/ (via rmarkdown)

─ Packages ───────────────────────────────────────────────────────────────────────────────────────────────────────────
 package        * version    date (UTC) lib source
 abind            1.4-5      2016-07-21 [1] RSPM (R 4.3.0)
 arrayhelpers     1.1-0      2020-02-04 [1] RSPM (R 4.3.0)
 backports        1.4.1      2021-12-13 [1] RSPM (R 4.3.0)
 base64enc        0.1-3      2015-07-28 [1] RSPM (R 4.3.0)
 bayesplot      * 1.10.0     2022-11-16 [1] RSPM (R 4.3.0)
 bit              4.0.5      2022-11-15 [1] RSPM (R 4.3.0)
 bit64            4.0.5      2020-08-30 [1] RSPM (R 4.3.0)
 bridgesampling   1.1-2      2021-04-16 [1] RSPM (R 4.3.0)
 brms           * 2.20.4     2023-09-25 [1] RSPM (R 4.3.0)
 Brobdingnag      1.2-9      2022-10-19 [1] RSPM (R 4.3.0)
 cachem           1.0.8      2023-05-01 [1] RSPM (R 4.3.0)
 callr            3.7.3      2022-11-02 [1] RSPM (R 4.3.0)
 checkmate        2.3.0      2023-10-25 [1] RSPM (R 4.3.0)
 cli              3.6.1      2023-03-23 [1] RSPM (R 4.3.0)
 cmdstanr       * 0.6.0.9000 2023-11-07 [1] Github (stan-dev/cmdstanr@a13c798)
 coda             0.19-4     2020-09-30 [1] RSPM (R 4.3.0)
 codetools        0.2-19     2023-02-01 [2] CRAN (R 4.3.1)
 colorspace       2.1-0      2023-01-23 [1] RSPM (R 4.3.0)
 colourpicker     1.3.0      2023-08-21 [1] RSPM (R 4.3.0)
 conflicted     * 1.2.0      2023-02-01 [1] RSPM (R 4.3.0)
 cowplot        * 1.1.1      2020-12-30 [1] RSPM (R 4.3.0)
 crayon           1.5.2      2022-09-29 [1] RSPM (R 4.3.0)
 crosstalk        1.2.0      2021-11-04 [1] RSPM (R 4.3.0)
 curl             5.1.0      2023-10-02 [1] RSPM (R 4.3.0)
 digest           0.6.33     2023-07-07 [1] RSPM (R 4.3.0)
 directlabels   * 2023.8.25  2023-09-01 [1] RSPM (R 4.3.0)
 distributional   0.3.2      2023-03-22 [1] RSPM (R 4.3.0)
 dplyr          * 1.1.3      2023-09-03 [1] RSPM (R 4.3.0)
 DT               0.30       2023-10-05 [1] RSPM (R 4.3.0)
 dygraphs         1.1.1.6    2018-07-11 [1] RSPM (R 4.3.0)
 ellipsis         0.3.2      2021-04-29 [1] RSPM (R 4.3.0)
 evaluate         0.22       2023-09-29 [1] RSPM (R 4.3.0)
 fansi            1.0.5      2023-10-08 [1] RSPM (R 4.3.0)
 farver           2.1.1      2022-07-06 [1] RSPM (R 4.3.0)
 fastmap          1.1.1      2023-02-24 [1] RSPM (R 4.3.0)
 forcats        * 1.0.0      2023-01-29 [1] RSPM (R 4.3.0)
 fs             * 1.6.3      2023-07-20 [1] RSPM (R 4.3.0)
 furrr          * 0.3.1      2022-08-15 [1] RSPM (R 4.3.0)
 future         * 1.33.0     2023-07-01 [1] RSPM (R 4.3.0)
 generics         0.1.3      2022-07-05 [1] RSPM (R 4.3.0)
 ggdist           3.3.0      2023-05-13 [1] RSPM (R 4.3.0)
 ggplot2        * 3.4.4      2023-10-12 [1] RSPM (R 4.3.0)
 globals          0.16.2     2022-11-21 [1] RSPM (R 4.3.0)
 glue           * 1.6.2      2022-02-24 [1] RSPM (R 4.3.0)
 gridExtra        2.3        2017-09-09 [1] RSPM (R 4.3.0)
 gtable           0.3.4      2023-08-21 [1] RSPM (R 4.3.0)
 gtools           3.9.4      2022-11-27 [1] RSPM (R 4.3.0)
 hms              1.1.3      2023-03-21 [1] RSPM (R 4.3.0)
 htmltools        0.5.6.1    2023-10-06 [1] RSPM (R 4.3.0)
 htmlwidgets      1.6.2      2023-03-17 [1] RSPM (R 4.3.0)
 httpuv           1.6.12     2023-10-23 [1] RSPM (R 4.3.0)
 igraph           1.5.1      2023-08-10 [1] RSPM (R 4.3.0)
 inline           0.3.19     2021-05-31 [1] RSPM (R 4.3.0)
 jsonlite         1.8.7      2023-06-29 [1] RSPM (R 4.3.0)
 knitr            1.44       2023-09-11 [1] RSPM (R 4.3.0)
 labeling         0.4.3      2023-08-29 [1] RSPM (R 4.3.0)
 later            1.3.1      2023-05-02 [1] RSPM (R 4.3.0)
 lattice          0.21-8     2023-04-05 [2] CRAN (R 4.3.1)
 lifecycle        1.0.3      2022-10-07 [1] RSPM (R 4.3.0)
 listenv          0.9.0      2022-12-16 [1] RSPM (R 4.3.0)
 loo              2.6.0      2023-03-31 [1] RSPM (R 4.3.0)
 lubridate      * 1.9.3      2023-09-27 [1] RSPM (R 4.3.0)
 magrittr       * 2.0.3      2022-03-30 [1] RSPM (R 4.3.0)
 markdown         1.11       2023-10-19 [1] RSPM (R 4.3.0)
 Matrix           1.5-4.1    2023-05-18 [2] CRAN (R 4.3.1)
 matrixStats      1.0.0      2023-06-02 [1] RSPM (R 4.3.0)
 memoise          2.0.1      2021-11-26 [1] RSPM (R 4.3.0)
 mime             0.12       2021-09-28 [1] RSPM (R 4.3.0)
 miniUI           0.1.1.1    2018-05-18 [1] RSPM (R 4.3.0)
 munsell          0.5.0      2018-06-12 [1] RSPM (R 4.3.0)
 mvtnorm          1.2-3      2023-08-25 [1] RSPM (R 4.3.0)
 nlme             3.1-162    2023-01-31 [2] CRAN (R 4.3.1)
 parallelly       1.36.0     2023-05-26 [1] RSPM (R 4.3.0)
 pillar           1.9.0      2023-03-22 [1] RSPM (R 4.3.0)
 pkgbuild         1.4.2      2023-06-26 [1] RSPM (R 4.3.0)
 pkgconfig        2.0.3      2019-09-22 [1] RSPM (R 4.3.0)
 plyr             1.8.9      2023-10-02 [1] RSPM (R 4.3.0)
 posterior      * 1.4.1      2023-03-14 [1] RSPM (R 4.3.0)
 prettyunits      1.2.0      2023-09-24 [1] RSPM (R 4.3.0)
 processx         3.8.2      2023-06-30 [1] RSPM (R 4.3.0)
 promises         1.2.1      2023-08-10 [1] RSPM (R 4.3.0)
 ps               1.7.5      2023-04-18 [1] RSPM (R 4.3.0)
 purrr          * 1.0.2      2023-08-10 [1] RSPM (R 4.3.0)
 quadprog         1.5-8      2019-11-20 [1] RSPM (R 4.3.0)
 QuickJSR         1.0.7      2023-10-15 [1] RSPM (R 4.3.0)
 R6               2.5.1      2021-08-19 [1] RSPM (R 4.3.0)
 Rcpp           * 1.0.11     2023-07-06 [1] RSPM (R 4.3.0)
 RcppParallel     5.1.7      2023-02-27 [1] RSPM (R 4.3.0)
 readr          * 2.1.4      2023-02-10 [1] RSPM (R 4.3.0)
 reshape2         1.4.4      2020-04-09 [1] RSPM (R 4.3.0)
 rlang          * 1.1.1      2023-04-28 [1] RSPM (R 4.3.0)
 rmarkdown        2.25       2023-09-18 [1] RSPM (R 4.3.0)
 rstan            2.32.3     2023-10-15 [1] RSPM (R 4.3.0)
 rstantools       2.3.1.1    2023-07-18 [1] RSPM (R 4.3.0)
 rstudioapi       0.15.0     2023-07-07 [1] RSPM (R 4.3.0)
 rsyslog        * 1.0.3      2023-05-08 [1] RSPM (R 4.3.0)
 scales         * 1.2.1      2022-08-20 [1] RSPM (R 4.3.0)
 sessioninfo      1.2.2      2021-12-06 [1] RSPM (R 4.3.0)
 shiny            1.7.5.1    2023-10-14 [1] RSPM (R 4.3.0)
 shinyjs          2.1.0      2021-12-23 [1] RSPM (R 4.3.0)
 shinystan        2.6.0      2022-03-03 [1] RSPM (R 4.3.0)
 shinythemes      1.2.0      2021-01-25 [1] RSPM (R 4.3.0)
 StanHeaders      2.26.28    2023-09-07 [1] RSPM (R 4.3.0)
 stringi          1.7.12     2023-01-11 [1] RSPM (R 4.3.0)
 stringr        * 1.5.0      2022-12-02 [1] RSPM (R 4.3.0)
 svUnit           1.0.6      2021-04-19 [1] RSPM (R 4.3.0)
 tensorA          0.36.2     2020-11-19 [1] RSPM (R 4.3.0)
 threejs          0.3.3      2020-01-21 [1] RSPM (R 4.3.0)
 tibble         * 3.2.1      2023-03-20 [1] RSPM (R 4.3.0)
 tidybayes      * 3.0.6      2023-08-12 [1] RSPM (R 4.3.0)
 tidyr          * 1.3.0      2023-01-24 [1] RSPM (R 4.3.0)
 tidyselect       1.2.0      2022-10-10 [1] RSPM (R 4.3.0)
 tidyverse      * 2.0.0      2023-02-22 [1] RSPM (R 4.3.0)
 timechange       0.2.0      2023-01-11 [1] RSPM (R 4.3.0)
 tzdb             0.4.0      2023-05-12 [1] RSPM (R 4.3.0)
 utf8             1.2.4      2023-10-22 [1] RSPM (R 4.3.0)
 V8               4.4.0      2023-10-09 [1] RSPM (R 4.3.0)
 vctrs            0.6.4      2023-10-12 [1] RSPM (R 4.3.0)
 vroom            1.6.4      2023-10-02 [1] RSPM (R 4.3.0)
 withr            2.5.1      2023-09-26 [1] RSPM (R 4.3.0)
 xfun             0.40       2023-08-09 [1] RSPM (R 4.3.0)
 xtable           1.8-4      2019-04-21 [1] RSPM (R 4.3.0)
 xts              0.13.1     2023-04-16 [1] RSPM (R 4.3.0)
 yaml             2.3.7      2023-01-23 [1] RSPM (R 4.3.0)
 zoo              1.8-12     2023-04-13 [1] RSPM (R 4.3.0)

 [1] /usr/local/lib/R/site-library
 [2] /usr/local/lib/R/library

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Code

options(width = 80L)